Is there any way to make edges corners become rounded?
Rounding the corners of the inset face cannot be done directly.

But you can extrude it and so have a "volume" that can be beveled.

 


Select the inset face
Extrude E
Go to edge select mode CtrTab
Select the edges
Bevel CtrlB (use mouse wheel or +/- from the numpad to choose the amount of bevel)
Once ok, remove the extrude face (select it then X then vertices)
Remake the original face: back to edge select mode CtrTab, then Alt select the border then F to make the face